AI 中文总结
该研究探讨三维随机定向曼哈顿格上随机游走的超扩散性，证明临界维度下扩散系数的Tauberian发散行为，解答了相关学者2018年提出的猜想。
AI 中文摘要
我们研究随机定向曼哈顿格上随机游走的超扩散行为，该格为d维整数格ℤᵈ，其中每条轴向线独立以等概率被赋予随机方向（正向或反向）。游走者采取最近邻步，随机选择一个轴并沿该轴的线所分配的方向移动，每个轴被选中的概率相等。我们证明，在临界维度d=3时，当时间t→∞，随机游走的扩散系数在Tauberian意义下以√log t发散，带有乘性修正项(log log t)^(±(2+ε))。这回答了Ledger、Tóth和Valkó（2018）提出的猜想。
英文摘要
We study the superdiffusive behavior of random walks on the randomly oriented Manhattan lattice, i.e., the $d$-dimensional integer lattice $\mathbb{Z}^d$ where each axis-aligned line is independently assigned a random direction (forward or backward) with equal probability. The walker takes nearest-neighbor steps, choosing an axis randomly and moving along the assigned direction of that axis's line, with equal probabilities for each axis. We show that, in the critical dimension $d=3$, the diffusion coefficient of the random walk diverges in the Tauberian sense as $\sqrt{\log t}$ with a multiplicative correction $(\log\log t)^{\pm(2+\varepsilon)}$ as time $t\to\infty$. This gives an answer to a conjecture by Ledger, Tóth and Valkó (2018).
Comments46 pages, 1 figure
